On Asymptotic Optimality of the Second Order in the Minimax Quickest Detection Problem of Drift Change for Brownian Motion

This paper deals with the minimax quickest detection problem of a drift change for the Brownian motion. The following minimax risks are studied: $C(T)=\inf_{\tau\in{\mathfrak{M}}_{T}}\sup_\thetaE_\theta(\tau-\theta\,|\,\tau\ge\theta)$ and $\overline{C}(T)=\inf_{\overline{\tau}\in\overline{\mathfrak{M}}_T}\sup_\thetaE_\theta(\overline{\tau}-\theta\,|\,\overline{\tau}\ge\theta)$, where ${\mathfrak{M}}_T$ is the set of stopping times $\tau$ such that $E_\infty\tau=T$ and ${\overline{\mathfrak{M}}}_T$ is the set of randomized stopping times ${\overline{\tau}}$ such that $E_\infty{\overline{\tau}}=T$. The goal of this paper is to obtain for these risks estimates from above and from below. Using these estimates we prove the existence of stopping times, which are asymptotically optimal of the first and second orders as $T\to\infty$ (for $C(T)$ and $\overline{C}(T)$, respectively).